Pure discrete spectrum and regular model sets on some non-unimodular substitution tilings

The equivalence between pure discrete spectrum and regular model sets on some non-unimodular substitution tilings is established. This will help to provide useful information about the cut-and-project scheme used in the description of quasiperiodic structures.


Introduction
There has been considerable success in studying the structure of tilings with pure discrete spectrum by setting them in the context of model sets (Baake & Moody, 2004;Baake et al., 2007;Strungaru, 2017;Akiyama et al., 2015). However, in general settings, the relation between pure discrete spectrum and model sets is not completely understood and the cut-andproject scheme is usually constructed with an abstract internal space (Baake & Moody, 2004;Strungaru, 2017). Thus it is not easy to understand this relation concretely and get information about the structure from the relation. The notion of inter model sets was introduced by Baake et al. (2007) and Lee & Moody (2006) and we know the equivalence between pure discrete spectrum and inter model sets in substitution tilings (Lee, 2007). But there are still some limitations in getting useful information about the cut-and-project scheme (CPS) because the internal space was constructed abstractly. What is the internal space concretely? There was some progress in this direction by Lee et al. (2018) and . However, these papers make various assumptions about substitution tilings such as the expansion map is diagonalizable, the eigenvalues of the expansion map should be algebraically conjugate, the multiplicity of the eigenvalues should be the same, and the expansion map is unimodular. From a long perspective, we aim to gradually eliminate assumptions one by one. As a first step, in this paper we eliminate the assumption of unimodularity.
Our work was inspired by an example of Baake et al. (1998), which offers a guide to what the internal space should be. We will look at this in Example 5.10. The present paper is an extension of the result of  in the sense that the unimodularity condition is removed, and the setting is quite similar.
There are various research works on non-unimodular substitution cases (Baker et al., 2006;Ei et al., 2006;Siegel, 2002) that study symbolic substitution sequences or their geometric substitution tilings in dimension 1. Our definition of non-unimodularity looks slightly different from that defined in those papers. However, if we restrict the substitution tilings to one dimension R, the two definitions are the same.
We have four basic assumptions about a primitive substitution tiling T on R d with an expansion map : (i) is diagonalizable.
(ii) All the eigenvalues of are algebraically conjugate.
(iii) All the eigenvalues of have the same multiplicity.
(iv) T is rigid [see (14) for the definition]. We call these assumptions DAMR. This paper relies heavily on the rigid structure of substitution tilings, and the rigidity property is only known under those assumptions (i), (ii), (iii) together with finite local complexity (Theorem 2.9). In Section 2, we review some definitions and known results that are going to be used in this paper. The main result of this paper shows the following: Theorem 1.1. Let T be a repetitive primitive substitution tiling on R d with a diagonalizable expansion map whose eigenvalues are algebraic conjugates with the same multiplicity and let T be rigid. If T has pure discrete spectrum, then control point set C j of each tile type is a regular model set in the CPS with an internal space which is a product of a Euclidean space and a profinite group, where C ¼ ðC j Þ 1 j is a control point set of T defined in (7) and the CPS is defined in (35).
In Section 3, we give an outline of the proof of this theorem in some simple case of substitution tilings with expansion map satisfying the DAMR assumptions defined above. In Section 4, we define an appropriate internal space and construct a CPS under the DAMR assumptions. Then we discuss the projected point sets E ;k of neighbourhood bases of a topology in the internal space. In Section 6, under the assumption of pure discrete spectrum of T , we look at how the projected point sets E ;k and the translation vector set Ä of the same types of tiles in T are related [see (8)]. Using the equivalent property 'algebraic coincidence' for pure discrete spectrum, we provide arguments to show that we actually have regular model sets.

Definitions and known results
We consider a primitive substitution tiling T on R d with expansion map satisfying the DAMR assumptions defined above. In this section, we recall some definitions and results that we are going to use in the later sections.

Tilings
We consider a set of types (or colours) f1; . . . ; g, which we fix once and for all. A tile in R d is defined as a pair T ¼ ðA; iÞ where A ¼ suppðTÞ (the support of T) is a compact set in R d , which is the closure of its interior, and i ¼ lðTÞ 2 f1; . . . ; g is the type of T. A tiling of R d is a set T of tiles such that R d ¼ [fsuppðTÞ : T 2 T g and distinct tiles have disjoint interiors.
Given a tiling T , a finite set of tiles of T is called a T -patch. Recall that a tiling T is said to be repetitive if the occurrence of every T -patch is relatively dense in space. We say that a tiling T has finite local complexity (FLC) if for every R > 0 there are only finitely many translational classes of T -patches whose support lies in some ball of radius R up to translations.

Delone j-sets
Recall that a Delone set is a relatively dense and uniformly discrete subset of R d . We say that Meyer, 1972;Lagarias, 1996;Moody, 1997). If Ã is a Delone -set and suppðKÞ is a Meyer set, we say that Ã is a Meyer -set.

Substitutions
We say that a linear map : for all x; y 2 R d under some metric d on R d compatible with the standard topology.
Definition 2.1. Let A ¼ fT 1 ; . . . ; T g be a finite set of tiles on R d such that T i ¼ ðA i ; iÞ; we will call them prototiles. Denote by P A the set of patches made of tiles each of which is a translate of one of the T i 's. We say that ! : A ! P A is a tilesubstitution (or simply substitution) with an expansive map if there exist finite sets D ij & R d for i; j , such that Here all sets in the right-hand side must have disjoint interiors; it is possible for some of the D ij to be empty.
The substitution (1) is extended to all translates of prototiles by !ðx þ T j Þ ¼ x þ !ðT j Þ, and to patches and tilings by !ðPÞ ¼ [f!ðTÞ : T 2 Pg. The substitution ! can be iterated, producing larger and larger patches ! k ðPÞ. A tiling T satisfying !ðT Þ ¼ T is called a fixed point of the tile-substitution or a substitution tiling with expansion map . It is known (and easy to see) (Solomyak, 1997) that one can always find a periodic point for ! in the tiling dynamical hull, i.e. ! N ðT Þ ¼ T for some N 2 N. In this case we use ! N in the place of ! to obtain a fixed point tiling. The substitution Â matrix S of the tile-substitution is defined by Sði; jÞ ¼ #D ij . We say that the substitution tiling T is primitive if there is an ' > 0 for which S ' has no zero entries, where S is the substitution matrix.
When there exists a monic polynomial PðxÞ over Z with the minimal degree satisfying PðÞ ¼ 0, we call the polynomial the minimal polynomial of over Z. We say that is unimodular if the minimal polynomial of over Z has constant term AE1; that is to say, the product of all roots of the minimal polynomial of is AE1. If the constant term in the minimal polynomial of is not AE1, then we say that is non-unimodular.
Note that for M 2 N, Definition 2.2. K ¼ ðÃ i Þ i is called a substitution Delone -set if Ã is a Delone -set and there exist an expansive map : R d ! R d and finite sets D ij for i; j such that where the unions on the right-hand side are disjoint.
Definition 2.3. For a substitution Delone -set K ¼ ðÃ i Þ i satisfying (4), define a matrix È ¼ ðÈ ij Þ i;j¼1 whose entries are finite (possibly empty) families of linear affine transformations on R d given by Thus ÈðKÞ ¼ K by definition. We say that È is a -set substitution. Let denote the substitution matrix corresponding to È.
Definition 2.4. (Mauduit, 1989.) An algebraic integer is a real Pisot number if it is greater than 1 and all its Galois conjugates are less than 1 in modulus, and a complex Pisot number if every Galois conjugate, except the complex conjugate , has modulus less than 1. A set of algebraic integers Â ¼ f 1 ; . . . ; r g is a Pisot family if for every 1 j r, every Galois conjugate of j , with jj ! 1, is contained in Â.
For r ¼ 1, with 1 real and j 1 j > 1, this reduces to j 1 j being a real Pisot number, and for r ¼ 2, with 1 non-real and j 1 j > 1, to 1 being a complex Pisot number.

Pure discrete spectrum and algebraic coincidence
Let X T be the collection of tilings on R d each of whose patches is a translate of a T -patch. In the case that T has FLC, there is a well known metric on the tilings: for a small > 0 two tilings S 1 ; S 2 are -close if S 1 and S 2 agree on the ball of radius À1 around the origin, after a translation of size less than (see Schlottmann, 2000;Radin & Wolff, 1992;Lee et al., 2003). Then where the closure is taken in the topology induced by the metric .
It is known that a dynamical system ðX T ; R d Þ with a primitive substitution tiling T always has a unique ergodic measure in the dynamical system ðX T ; R d Þ (see Solomyak, 1997;Lee et al., 2003). We consider the associated group of unitary operators fT x g x2R d on L 2 ðX T ; Þ: Every g 2 L 2 ðX T ; Þ defines a function on R d by x 7 ! hT x g; gi. This function is positive definite on R d , so its Fourier transform is a positive measure g on R d called the spectral measure corresponding to g. The dynamical system ðX T ; ; R d Þ is said to have pure discrete spectrum if g is pure point for every g 2 L 2 ðX T ; Þ. We also say that T has pure discrete spectrum if the dynamical system ðX T ; ; R d Þ has pure discrete spectrum.
The notion of pure discrete spectrum of the dynamical system is quite closely connected wtih the notion of algebraic coincidence in Definition 2.6. For this we start by introducing control points. There is a standard way to choose distinguished points in the tiles of a primitive substitution tiling so that they form a -invariant Delone -set. They are called control points (Thurston, 1989;Praggastis, 1999), which are defined below.
Definition 2.5. Let T be a primitive substitution tiling with an expansion map . For every T -tile T, we choose a tile ðTÞ in the patch !ðTÞ; for all tiles of the same type in T , we choose ðTÞ with the same relative position [i.e. if S ¼ x þ T for some two tiles S; T 2 T then ðSÞ ¼ x þ ðTÞ]. This defines a map : T ! T called the tile map. Then we define the control point for a tile T 2 T by The control points satisfy the following: (a) T 0 = T þ cðT 0 Þ À cðTÞ, for any tiles T; T 0 of the same type; (b) ðcðTÞÞ ¼ cððTÞÞ, for T 2 T . Let be a set of control points of the tiling T in R d . Let us denote For tiles of any tiling S 2 X T , the control points have the same relative position as in T -tiles. The choice of control points is non-unique, but there are only finitely many possibilities, determined by the choice of the tile map. Let Since the substitution tiling T is primitive, it is possible to assume that the substitution matrix S is positive taking S k if necessary. So we consider a tile map with the property that for every T 2 T , the tile ðTÞ has the same tile type in T . That is to say, for every T; S 2 T , ðTÞ ¼ x þ ðSÞ, where ðTÞ; ðSÞ 2 T and x 2 Ä. Then for any T; S 2 T , cððTÞÞ À cððSÞÞ 2 Ä: In order to have 0 2 C j for some j and C j & Ä, we define the tile map as follows. It is known that there exists a finite generating patch P for which lim n!1 ! n ðPÞ ¼ T (Lagarias & Wang, 2003). Although it was defined there for primitive substitution point sets, it is easy to see that the same property holds for primitive substitution tilings. We call the finite patch P the generating tile set. When we apply the substitution infinitely many times to the generating tile set P, we obtain the whole substitution tiling. So there exists n 2 N such that the nth iteration of the substitution to the generating tile set covers the origin. We choose a tile R in a patch ! n ðPÞ which contains the origin, where R ¼ a þ T j for some 1 j . Then there exists a fixed tile S 2 P such that R 2 ! n ðSÞ. Replacing the substitution ! by ! n , we can define a tile map so that ðTÞ is a j-type tile in ! n ðTÞ if T 2 T with T 6 ¼ S ðSÞ ¼ R:

&
Then 0 2 C j by the definition of the control point sets and so C j & Ä. Since ðcðTÞÞ ¼ cððTÞÞ for any T 2 T , ðC i Þ & C j for any i : Definition 2.6. (Lee, 2007.) Let T be a primitive substitution tiling on R d with an expansive map and let C ¼ ðC i Þ i be a corresponding control point set. We say that C admits an algebraic coincidence if there exists M 2 Z þ and 2 C i for some 1 i such that Note that if the algebraic coincidence is assumed, then for some k 2 Z þ , Theorem 2.7. [Theorem 3.13 (Lee, 2007), Theorem 2.6 (Lee, .] Let T be a primitive substitution tiling on R d with an expansive map and C ¼ ðC i Þ i be a control point set of T . Suppose that all the eigenvalues of are algebraic integers. Then T has pure discrete spectrum if and only if C admits an algebraic coincidence.

CPS
We use a standard definition for a CPS and model sets (see Baake & Grimm, 2013). For convenience, we give the definition for our setting.
Definition 2.8. A CPS consists of a collection of spaces and mappings as follows: where R d is a real Euclidean space, H is a locally compact Abelian group, 1 and 2 are the canonical projections, Here the set V is called a window of^ðVÞ. A subset Ã of R d is called a model set if Ã can be of the form^ðWÞ, where W & H has non-empty interior and compact closure in the setting of the CPS in (12). The model set Ã is regular if the boundary of W @W ¼ W \ W is of (Haar) measure 0. We say that K ¼ ðÃ i Þ i is a model -set (respectively, regular model -set) if each Ã i is a model set (respectively, regular model set) with respect to the same CPS.

Rigid structure on substitution tilings
The structure of a module generated by the control points is known only for the diagonalizable case for whose eigenvalues are algebraically conjugate with the same multiplicity given by Lee & Solomyak (2012). We need to use the structure of the module in the subsequent sections. Thus we will have the same assumptions.
Let J be the multiplicity of each eigenvalue of and assume that the number of distinct eigenvalues of is m. For 1 j J, we define a j 2 R d such that for each 1 k d, We recall the following theorem for the module structure of the control point sets. Although the theorem is not explicitly stated by Lee & Solomyak (2012), it can be read off from their Theorem 4.1 and Lemma 6.1.
Theorem 2.9. (Lee & Solomyak, 2012.) Let T be a repetitive primitive substitution tiling on R d with an expansion map .
Assume that T has FLC, is diagonalizable, and all the eigenvalues of are algebraically conjugate with the same multiplicity J. Then there exists a linear isomorphism where a j , 1 j J, are given as (13) and a 1 ; . . . ; a J 2 ðsupp CðT ÞÞ.
Note here that a 1 ; . . . ; a J are linearly independent over Z½. A tiling T is said to be rigid if T satisfies the result of Theorem 2.9; that is to say, there exists a linear isomorphism where a j , 1 j J, are given as (13).
As an example of a substitution tiling with the rigidity property, let us look at the Frank-Robinson substitution tiling (Frank & Robinson, 2006) (Fig. 1).
Take the tile-substutition where b is the largest root of x 2 À x À 3 ¼ 0 and Then it gives a primitive substitution tiling T . Note that b is not a Pisot number. It was shown by Frank & Robinson (2006) that 3. Outline of the proof of Theorem 1.1 We provide a brief outline of the proof of Theorem 1.1 for the simpler case of repetitive primitive substitution tilings T on R with an expansion factor ( ¼ ): (a) is non-unimodular, (b) is a real Pisot number which is not an integer, (c) T has FLC, (d) T has pure discrete spectrum. Let PðxÞ be the minimal polynomial of over Z for which PðxÞ ¼ a 0 þ a 1 x þ Á Á Á þ a n x n . Let ; 2 ; . . . ; n be all the roots of the equation PðxÞ ¼ 0, where the absolute values of 2 ; . . . ; n are all less than 1. Using the rigidity of Theorem 2.9, we get up to an isomorphism supp CðT Þ & Z½: Using the algebraic conjugates 2 ; . . . ; n of whose absolute values are less than 1, we consider a Euclidean space R nÀ1 and the map For the case of non-unimodular , we construct a profinite group below. We remark that if is unimodular, then the profinite group is trivial so that Theorem 1.1 can be covered by the work of . Let L ¼ Z½. From the non-unimodularity of , a 0 6 ¼ AE1. So L 6 L. Note that f1; ; . . . ; nÀ1 g is a basis of L as a free Z-module. Consider the map This gives an isomorphism of the Z-module between L and Z n . Let The Frank-Robinson tiling substitution.
Since Z n embeds in Z n M À , we can identify Z n with its image in Z n M À . Consider the following map: Now we construct a CPS whose physical space is R and Under the assumption of pure discrete spectrum of T , we know that an algebraic coincidence occurs by Theorem 2.7. So there exist M 2 Z þ and 2 C i for some 1 i such that where Ä is the set of translational vectors which translate a tile to the same type of tile in T as given in (8). Notice that B ð0Þ Â M k Z n M À is a basis element in the locally compact abelian group R nÀ1 Â Z n M À where B ð0Þ is a ball of radius around 0 in R nÀ1 . We let E ;k ¼^ðB ð0Þ Â M k Z n M À Þ be the projected point set in R coming from a window B ð0Þ Â M k Z n M À . It is important to understand the relation between E ;k and Ä. We discuss this in Section 4.2 (see also Lee et al., 2018;. From this relation, together with algebraic coincidence, we can view the control point set of T as a model set. Using Keesling's argument (Keesling, 1999), we show that the control point set of T is actually a regular model set.

Construction of a CPS
We aim to prove that the structure of pure discrete spectrum in a substitution tiling can be described by a regular model set which comes from a CPS with the internal space that is a product of a Euclidean space and a profinite group. From Lee & Solomyak (2019), under the assumption of pure discrete spectrum, the control point set of the substitution tiling has the Meyer property and so has FLC. In general settings which are not substitution tilings, it is hard to expect that pure discrete spectrum implies neither the Meyer property nor FLC (Lee, Lenz et al., 2020).
The setting that we consider here is a primitive substitution tiling T on R d with an expansion map which satisfies the DAMR assumptions. Changing the tile substitution if neces-sary, we can assume that is a diagonal matrix without loss of generality.
Under the assumption of DAMR, it is also known from Lee & Solomyak (2012, 2019 that the control point set of the substitution tiling has the Meyer property if and only if the eigenvalues of form a Pisot family. In our setting, there is no algebraic conjugate with jj ¼ 1 for the eigenvalues of , since is an expansion map. It is known that if is an expansion map of a primitive substitution tiling with FLC, every eigenvalue of is an algebraic integer (Kenyon & Solomyak, 2010;Kwapisz, 2016). Even for non-FLC cases, we know from the rigidity that the control point set lies in a finitely generated free abelian group L which spans R d and L & L. So all the eigenvalues of are algebraic integers [Lemma 4.1 of Lee & Solomyak (2008)].
In the case of non-unimodular substitution tilings, there are two parts of spaces for the internal space of a CPS. One is a Euclidean part and the other is a profinite group part. We describe them below.

An internal space for a CPS
4.1.1. Euclidean part for the internal space. In this subsection, we assume that there exists at least one algebraic conjugate whose absolute value is less than 1, which is different from the eigenvalues of . In the case of unimodular , we can observe that there always exists such an algebraic conjugate. But in the case of non-unimodular , it is possible not to have an algebraic conjugate whose absolute value is less than 1. For example, let us consider an expansion map Then the minimal polynomial of is x 2 À 6x þ 7 ¼ 0, which means that is non-unimodular. If there exists no other algebraic conjugate of the eigenvalues of whose absolute value is less than 1, one can skip this subsection and go to the next Section 4.1.2. Recall that J is the multiplicity of the eigenvalues of , d is the dimension of the space R d , m is the number of distinct eigenvalues of and d ¼ mJ. We can write where A k is a real 1 Â 1 matrix for 1 k s, a real 2 Â 2 matrix of the form a k Àb k b k a k ! for s þ 1 k s þ t with s; t 2 Z !0 and m ¼ s þ 2t. Here O is the m Â m zero matrix and 1 j J. Then the eigenvalues of are We assume that the minimal polynomial of over Z has e real roots and f pairs of complex conjugate roots. Since the minimal polynomial of has the characteristic polynomial of as a divisor, we can consider the roots of the minimal polynomial of over Z in the following order: 1 ; . . . ; s ; sþ1 ; . . . : e ; eþ1 ; eþ1 ; . . . ; eþt ; eþt ; eþtþ1 ; eþtþ1 ; . . . ; eþf ; eþf : We now consider a Euclidean space whose dimension is n À m, whose number corresponds to the number of the other roots of the minimal polynomial of which are not the eigenvalues of . Let For 1 j J, define a ðn À mÞ Â ðn À mÞ matrix where A sþg is a real 1 Â 1 matrix with the value sþg for 1 g e À s, and A eþtþh is a real 2 Â 2 matrix of the form a eþtþh Àb eþtþh b eþtþh a eþtþh ! for 1 h f À t [see  for more details]. The matrix D j operates on the space H j . Notice that and have the same minimal polynomial over Z, since is the diagonal matrix containing J copies of .
Let us consider now the following embeddings: where P j 2 Z½x, a j is as in (13), b j :¼ ð1; . . . ; 1Þ 2 H j and 1 j J. Note that É j ðxÞ ¼ D j É j ðxÞ for any x 2 Z½a j : Note that the minimal polynomial of is monic, since the eigenvalues of are all algebraic integers. So L & L and fa 1 ; . . . ; nÀ1 a 1 ; . . . ; a J ; . . . ; nÀ1 a J g ð 21Þ is a basis of L as a free Z-module. Now, we can define the map Since a 1 ; . . . ; a J are linearly independent over Z½, the map É 0 is well defined. Thus É 0 ðxÞ ¼ DÉ 0 ðxÞ where is a block diagonal ðn À mÞJ Â ðn À mÞJ matrix in which D j is an ðn À mÞ Â ðn À mÞ matrix, 1 j J, and 4.1.2. Profinite group part for the internal space. To make the notation short, denote the basis of L given in (21) by v 1 ; . . . ; v n ; . . . ; v ðnÀ1ÞJþ1 ; . . . ; v nJ . Consider a Z-module isomorphism between L and Z nJ : L ! Z nJ ; ðvÞ ¼ ðc 1 ; . . . ; c n ; . . . ; c ðnÀ1ÞJþ1 ; . . . ; c nJ Þ where Consider the ðd Â nJÞ matrix: Since L spans R d over R, the rank of N is d.
Notice that in a special case of J ¼ 1, i.e. L ¼ Z½a 1 , M is the companion matrix of the minimal polynomial of over Z. Then Note that for any k 2 N, and Notice also that for any v 2 L and for any k 2 N, Lemma 4.1. Any eigenvalue of with multiplicity J becomes also the eigenvalue of M with the same multiplicity J. Furthermore the minimal polynomial of over Z is the same as the minimal polynomial of M over Z.
Proof. Let be an eigenvalue of with multiplicity J. Since T and have the same eigenvalues, is an eigenvalue of T . Let x be the corresponding eigenvector of T . Then Since x is nonzero, N T x is nonzero and so is an eigenvalue of M T . Thus the eigenvalue of becomes also an eigenvalue of research papers M. Since is a diagonal matrix, there are dð¼ mJÞ independent eigenvectors. The images of these vectors under N T are the eigenvectors of M T and linearly independent. Since all the eigenvalues of are algebraically conjugate with the same multiplicity J, all the eigenvalues of are also eigenvalues of M T with the same multiplicity J. Thus we note that the set of the eigenvalues of M consists of all the eigenvalues of and all the other algebraic conjugates of them which are not the eigenvalues of , and the multiplicity of all the eigenvalues of M is J.
Since is a diagonal matrix and all the eigenvalues of are algebraic integers, there exists a minimal polynomial of over Z. Since M is an integer matrix, there exists a minimal polynomial of M over Z as well. Let PðxÞ be the minimal polynomial of over Z so that PðÞ ¼ 0 where PðxÞ = x k þ a kÀ1 x kÀ1 þ Á Á Á þ a 1 x þ a 0 , a i 2 Z, and i 2 f0; 1; . . . ; k À 1g. Then using (30), for any v 2 L, From (31), PðMÞ is a zero matrix. On the other hand, we can observe that if PðxÞ is the minimal polynomial of M over Z, then PðÞ is a zero matrix as well. Thus the minimal polynomial of over Z is the same as the minimal polynomial of M over Z.

&
We can observe this property of Lemma 4.1 concretely with Example 5.10.
Let us consider the case that is non-unimodular, i.e. L & L but L 6 ¼ L. Let us denote Z nJ by L which is a lattice in R nJ . Then ML & L but ML 6 ¼ L. We define the M-adic space which is an inverse limit space of L=M k L with k 2 N. Note that M : L ! L is an injective homomorphism. Observe that ½L : ML is non-trivial and finite. We have an inverse limit of an inverse system of discrete finite groups, which is a profinite group. Note that L M can be supplied with the usual topology of a profinite group. Note that for any element x = ðx 1 þ ML; Note that L M contains a canonical copy of L via the mapping We can observe that Note that \ 1 k¼0 M k L ¼ f0g. So we can conclude that the mapping x 7 ! fx mod M k Lg k embeds L in L M . We identify L with its image in L M . Note that L M is the closure of L with respect to the topology induced by the metric d.
In the unimodularity case of , L ¼ L and so ML ¼ L. Thus L M is trivial.

Concrete construction of a CPS
We construct a CPS taking R d as a physical space and H Â L M as an internal space. We will consider this construction dividing into three cases as given in the following remark. The following construction of a CPS has already appeared in the work of Minervino & Thuswaldner (2014) in the case of d ¼ 1. Here we construct a CPS for the case of d ! 1.
Remark 4.2. For an expansion map , there are three cases.
(i) If is unimodular, there exists at least one algebraic conjugate other than the eigenvalues of for which jj < 1. Then the map in (33) is a trivial map and the internal space is constructed mainly by the Euclidean space discussed in Section 4.1.1.
(ii) If is non-unimodular and there exists no other algebraic conjugate of the eigenvalues of whose absolute value is less than 1, then H is a trivial group and the internal space is constructed exclusively by the profinite group (32) defined in Section 4.1.2.
(iii) If is non-unimodular and there exist algebraic conjugates ('s) other than the eigenvalues of for which jj < 1, then the internal space is a product of the Euclidean space in Section 4.1.1 and the profinite group in Section 4.1.2.

Let us define
where is defined as in (24). Let us construct a CPS: where 1 and 2 are canonical projections, It is easy to see that 1 j e L L is injective. We shall show that 2 ð e L LÞ is dense in H Â L M and e L L is a lattice in R d Â H Â L M in Lemmas 4.3 and 4.4. We note that 2 j e L L is injective, since É is injective. Since commutes with the isomorphism in Theorem 2.9, we may identify the control point set CðT Þ ¼ ðC i Þ i with its isomorphic image. Thus from Theorem 2.9, where a 1 ; . . . ; a J 2 CðT Þ and supp CðT Þ ¼ [ i C i . Note that for any k 2 N and 1 j J, k a j 2 CðT Þ by the definition of the tile-map. So we can note that Proof. For the case (i) of Remark 4.2, L M is trivial. So the statement of the lemma follows from Lemma 3.2 of . For the case (ii) of Remark 4.2, H is trivial. Note that the matrix M in (26) is a d Â d integer matrix and L is a lattice in R d . So e L L is a discrete subgroup of R d Â L M with respect to the product topology. Note that Thus the statement of the lemma follows.
For the case (iii) of Remark 4.2, let L 0 = fðx; É 0 ðxÞÞ : x 2 Lg & R d Â H. In Lemma 3.2 of , we notice that the unimodularity property is used only in observing that H is not trivial in that paper. So by the same argument as Lemma 3.2 of Lee, , we obtain that L 0 is a lattice in R d Â H. This means that L 0 is a discrete subgroup such that ðR d Â HÞ=L 0 is compact. Notice that e L L is still a discrete subgroup in R d Â H Â L M . Furthermore, ðR d Â H Â L M Þ= e L L is compact. In fact, note that where C 1 and C 2 are compact sets in R d and H, respectively. Then Proof. For the case (i) of Remark 4.2, L M is trivial. So the statement of the lemma follows from Lemma 3.2 of .
For the case (ii) of Remark 4.2, H is trivial. Note that { ðLÞ ¼ L and L is dense in L M . Thus 2 ð e L LÞ is dense in L M .
Let us consider the case (iii) of Remark 4.2. It is known from Hence Éð À1 ðz 0 þ M ' LÞÞ \ ðV Â ðz þ M k L M ÞÞ 6 ¼ ;: Now that we have proved that (35) is a CPS, we would like to introduce a special projected set E ;k which will appear in the proof of the main result in Section 5. For > 0 and k 2 Z !0 , we define where B H ð0Þ is an open ball around 0 with a radius in H and In the following lemma, we find an adequate window for a set n E ;k and note that E ;k is a Meyer set.
The third equivalence comes from (34) and the fourth equivalence comes from (30). Thus In the unimodularity case of , L M is trivial and L ¼ L. So the last equality (39) follows. In the non-unimodularity case of , {ððQðÞaÞÞ 2 M kþ1 L M implies QðÞa 2 kþ1 L. Since kþ1 L & L, QðÞa 2 L. This shows the last equality (39). Hence for any n 2 N, Since (35) is a CPS, B H ð0Þ is bounded, and L M is compact, B H ð0Þ Â M k L M has a non-empty interior and compact closure, E ;k is a model set for each > 0 and k 2 Z !0 . It is given by Moody (1997) and Meyer (1972) that a model set is a Meyer set. Thus E ;k forms a Meyer set for each > 0 and

Main result
Recall that we consider a primitive substitution tiling T on R d with a diagonal expansion map whose eigenvalues are algebraically conjugate with the same multiplicity J and T is rigid.
Under the assumption of the rigidity of T , the pure discrete spectrum of T implies that the set of eigenvalues of forms a Pisot family [Lemma 5.1 (Lee & Solomyak, 2012)]. Recall that where CðT Þ ¼ ðC i Þ i is a control point set of T .
Proof. Notice that the setting for T fulfils the conditions to use Lemma 4.5 of Lee & Solomyak (2008). So from this lemma, for any y 2 Ä, y ¼ P N n¼0 n x n ; where x n 2 U and U is a finite subset in L: Recall that is an expansive map and satisfies the Pisot family condition. If there exists at least one algebraic conjugate other than the eigenvalues of for which jj < 1, Assume that T has pure discrete spectrum. Then for any y 2 L, there exists ' ¼ 'ðyÞ 2 N such that ' y 2 Ä.
Proof. Note from (36) that for any k 2 N and a j 2 fa 1 ; . . . ; a J g, k a j is contained in Ä. Recall that L ¼ Z½a 1 þ Á Á Á þ Z½a J . From (10) and (36), So for any y 2 L, y is a linear combination of a 1 ; a 1 ; . . . ; nÀ1 a 1 ; . . . ; a J ; a J ; . . . ; nÀ1 a J over Z. Applying (11) many times if necessary, we get that for any y 2 L, ' y 2 Ä for some ' ¼ 'ðyÞ 2 N. & Proposition 5.3. Let T be a primitive substitution tiling on R d with an expansion map . Under the assumption of the existence of the CPS (35), if T has pure discrete spectrum, then for any given > 0, there exists K 2 N such that Proof. Note that E ;0 is a Meyer set and Ä & E ;0 for some > 0. Since Ä is relatively dense, for any x 2 E ;0 , there exists r > 0 such that Ä \ B R d r ðxÞ 6 ¼ ;. It is important to note that from the Meyer property of E ;0 , the point set configurations fÄ \ B R d r ðxÞ : x 2 E ;0 g are finite up to translations. Let and x 2 E ;0 g: Then F & L and F is a finite set. Thus for any x 2 E ;0 , From Lemma 5.2, for any y 2 L, there exists ' ¼ 'ðyÞ 2 N such that ' y 2 Ä. Since T has pure discrete spectrum and so T admits algebraic coincidence, by (11) there exists K 1 2 N such that Applying the inclusion (43) finitely many times, we obtain that there exists K 0 2 N such that K 0 F & Ä. Hence together with (42), there exists K 2 N such that & Proposition 5.4. Let T be a primitive substitution tiling on R d with a diagonalizable expansion map whose eigenvalues are algebraic conjugates with the same multiplicity and let T be rigid. Let È be the corresponding -set substitution of T (see Definition 2.3). Suppose that for some 0 2 C j , j and N 2 Z þ . Then each point set Proof. For each i and z 2 C i , there exist n 2 N and j for which z ¼ f ð0Þ with some f 2 ðÈ n Þ ij : By Theorem 2.7 and Proposition 5.3, there exists K 2 N such that K E ;0 & Ä. Thus where N z 2 N and N z depends on z. Let where z Ã ¼ ÉðzÞ. Then for any i In (47), we assume that we have taken the minimal number N z 2 N so that U i defined by using N z À 1 does not satisfy  (23) and (26), and a Ã ¼ ÉðaÞ. If there is no confusion, we will use the same notation f Ã for the extended map. Note that, by the Pisot family condition on , if there exists at least one algebraic conjugate other than the eigenvalues of for which jj < 1, there exists some c < 1 such that jDxj c Á jxj for any x 2 H. Furthermore, from (33) By the same argument as in Section 3 of Lee & Moody (2001), the -set substitution È induces a multi-component iterated function system on L M . Thus the -set substitution È determines a multi-component iterated function system È Ã on H Â L M and f Ã is a contraction on H Â L M . Let SðÈ Ã Þ ¼ ðcardðÈ Ã ij ÞÞ ij be a substitution matrix corresponding to È Ã . Defining the compact subsets W i ¼ ÉðC i Þ for each 1 i and using (5) and the continuity of the mappings, we have This shows that W 1 ; . . . ; W are the unique attractor of È Ã .
Lemma 5.5. Let where z Ã ¼ ÉðzÞ, as obtained in (47) with the minimal number N z satisfying (48). For any j and any K 2 N, we have Proof. For any i; j , ðÈ K Þ ij ðC j Þ & C i . Recall that So for any f 2 ððÈÞ K Þ ij , f ðC j Þ & C i . Thus for any f Ã 2 ððÈ Ã Þ K Þ ij , The following proposition shows that the Haar measure of @U i is zero for each 1 i . This is proved using Keesling's argument (Keesling, 1999).
Proposition 5.6. Let T be a primitive substitution tiling on R d with a diagonalizable expansion map whose eigenvalues are algebraic conjugates with the same multiplicity and let T be rigid. Let È be the corresponding -set substitution of T (see Definition 2.3). If where 0 2 C j , j and N 2 Z þ , then each model set C j , 1 j , has a window with boundary measure zero in the internal space H Â L M of CPS (35). (46). From the assumption of (52), we first note that fulfils the Pisot family condition from Theorem 2.7 and Lemma 5.1 of Lee & Solomyak (2012 where is a Haar meaure in H, is a Haar measure in L M , ¼ Â . Note that jdetDj=jdetMj < 1. In particular,
Then for any C 2 X C , there exists ðÀs; ÀhÞ 2 R d Â ðH Â L M Þ so that From the assumption of pure discrete spectrum and Remark 5.5 of Lee, Akiyama & Lee (2020), we can observe that the condition (52) is fulfilled in the following theorem.
Theorem 5.8. Let T be a repetitive primitive substitution tiling on R d with a diagonalizable expansion map whose eigenvalues are algebraic conjugates with the same multiplicity and let T be rigid. If T has pure discrete spectrum, then each control point set C j , 1 j , is a regular model set in CPS (35) with an internal space which is a product of a Euclidean space and a profinite group.
Proof. Through Section 4.1, we can construct the CPS (35) whose internal space is a product of a Euclidean space and a profinite group. Since T has pure discrete spectrum and is repetitive, we can find a substitution tiling S in X T such that where 0 2 ðC S Þ j , j and N 2 Z þ . From Propositions 5.3, 5.6 and 5.7, the statement of the theorem follows.
Corollary 5.9. Let T be a repetitive primitive substitution tiling on R d with a diagonalizable expansion map whose eigenvalues are algebraic conjugates with the same multiplicity and T be rigid. Then T has pure discrete spectrum if and only if each control point set C j , 1 j , is a regular model set in CPS (35) with an internal space which is a product of a Euclidean space and a profinite group.
Proof. It is known that regular model sets have pure discrete spectrum in quite a general setting (Schlottmann, 2000). Together with Theorem 5.8, we obtain the equivalence between pure discrete spectrum and a regular model set in substitution tilings. & Now let us look at an example given by Baake et al. (1998).
Example 5.10. We look at the example of non-unimodular substitution tiling which is studied by Baake et al. (1998). This example is proven to be a regular model set in the setting of a CPS constructed by Baake et al. (1998). This has also been considered by , but it could only be described as a model set, not a regular model set. Here in the setting of CPS (35), we show that this example gives a regular model set. The substitution matrix of the primitive two-letter substitution a ! aab b ! abab has the Perron-Frobenius eigenvalue :¼ 2 þ 2 1=2 which is a Pisot number. A geometric substitution tiling arising from this substitution can be obtained by replacing symbols a and b in this sequence by the intervals of length 'ðaÞ ¼ 1 and 'ðbÞ ¼ 2 1=2 . Then we have the following tile-substitution ! !ðT a Þ ¼ fT a ; 1 þ T a ; 2 þ T b g; where T a ¼ ð½0; 1; aÞ and T b ¼ ð½0; 2 1=2 ; bÞ. Since x 2 À 4x þ 2 = 0 is the minimal polynomial of over Q and the constant term of the polynomial is 2, the expansion factor is nonunimodular. Then we can construct a repetitive substitution tiling T using the substitution !.
From Theorem 2.9, we know that the control point set CðT Þ fulfils where PðxÞ 2 Z½x, ¼ 2 À 2 1=2 , and L M is a M-adic space. Since this substitution tiling is known to have pure discrete spectrum (see Baake et al., 1998), it admits an algebraic coincidence. By Proposition 4.4 of Lee (2007) and rewriting the substitution, if necessary, we know that there exists a substitution tiling S 2 X T such that C ¼ ðÀ i Þ i :¼ CðSÞ and N Ä & À j for which 0 2 À j ; j and N 2 Z þ : Then, by the same argument as in Proposition 5.4, research papers where N z depends on z and E z ;0 ¼^ðB z ð0Þ Â L M Þ for some ball B z ð0Þ of radius z around 0 in R. Let From Proposition 5.7, we can observe that the pure discrete spectrum of T gives a model set with an open and precompact window in the internal space R Â L M for the control point set CðT Þ. From Proposition 5.6, the measures of the boundaries of the windows are all zero. Now let us look at another example of a constant-length substitution tiling in R. This example shows that it is important to start with a control point set satisfying the containment (10).
Example 5.11. Consider a two-letter substitution defined as follows: a ! aba b ! bab: The expansion factor 3 and each prototile can be taken as a unit interval. Starting from bja, we can expand b to the lefthand side and a to the right-hand side, applying the substitution infinite times. Then we get the following bi-infinite sequence: We consider two prototiles T a and T b each of which corresponds to the letter a and the letter b. Following the sequence (59), we replace each letter by the corresponding prototile and obtain a substitution tiling T which is fixed under the substitution As a representative point of each tile, if one takes the left end of each interval in the tiling, one gets two point sets Ã a and Ã b such that Ã a ¼ 2Z and Ã b ¼ 1 þ 2Z. Since Ã a [ Ã b ¼ Z, we can take L ¼ Z. Notice in this case that the Euclidean part for the internal space is trivial and the profinite group is Notice that there does not exist n 2 N such that x þ 3 n Z 2Z for some x 2 Z: This means that neither Ã a nor Ã b can be described as a model set projected from a window whose interior is non-empty in Z 3 . However the substitution tiling T has pure discrete spectrum, since it is a periodic structure. The problem here is that the control point set K ¼ ðÃ i Þ i2fa;bg is not taken to satisfy the containment (10).
On the other hand, if we take the tile map : T ! T for which ðTÞ ¼ 3x þ T a and ðSÞ ¼ 3y þ 1 þ T a ; where T ¼ x þ T a and S ¼ y þ T b 2 T with x 2 2Z and y 2 1 þ 2Z, then the control point set CðT Þ ¼ ðC i Þ i2fa;bg is C a ¼ 2Z and satisfying the containment (10), and the profinite group is ¼^ð 4 3 þ 3Á L 3 Þ: Therefore CðT Þ can be described as a model set.

Further study
In this paper, the rigid structure property of substitution tilings is used to make a connection from pure discrete spectrum to regular model sets, especially to compute the boundary measure of windows. So far, the rigid structure property is known for substitution tilings whose expansion maps (Q) are diagonalizable and the eigenvalues of Q are algebraically conjugate with the same multiplicity (Lee & Solomyak, 2012). Thus it would be useful to know some rigid structure for more general settings. If the rigidity property is precisely known for general substitution tilings, it is expected that we will be able to find the connection from pure discrete spectrum to regular model sets.